I certainly don't know most of the details of Ptolemy's system, but I
think in essence it was a system that gave fairly good accuracy for the angular
positions of the sun and planets relative to the orientation of the earth
(apparently good enough to be consistent with all the ancient data, I would
guess all before that of Tycho Brahe, though I am not sure). The ancients had
no way of getting accurately the radial distances to these objects, or, I
presume, even the relative distances to them (and hence Ptolemy's system could
give relative distances to the sun and Venus that would be inconsistent with
the phases of Venus as seen once telescopes were invented). I have the
impression that the epicycles were an ancient analogue of terms in a Fourier
series and were an excellent method, though complicated, of getting a very good
approximation (though I have no idea whether any of the ancients considered the
possibility that one might need an infinite number of terms in the series to
get complete accuracy, or of the possibility, I would guess true for the
epicycle series if one were trying to model the perturbations caused by the
gravitational interactions between the planets, that the series would not
converge to the correct answer or maybe not even converge at all).
Now my point is that since only relative angles were being measured,
the absolute position, orientation, linear velocity, and angular velocity of
any single entity in the system was immaterial to the testable predictions.
(In modern terms we might say they were analogous to gauge degrees of freedom
that had no relevance to the prediction of relative angles.) Thus it was
perfectly consistent with the model to set the earth at the center and have it
be nonrotating. As far as predicting the angles of the sun and planet relative
to the orientation of the earth (i.e, the altitude and azimuth of the sun and
planets as seen from Greenwich), Ptolemy's system would be in good agreement
with a modern system using an accelerated, rotating coordinate system in which
the earth is at the center and at rest (provided that we exclude high-precision
modern data). Thus with a round earth, Ptolemy's system would have given the
same "missing day" for the voyagers as a system in which the earth is taken to
be rotating and/or revolving around the sun. In other words, the complication
of Ptolemy's system that Glenn alludes to is irrelevant to the explanation of
the "missing day."
Even today one could use a system of accelerated, rotating coordinates
and keep the earth nonrotating at the center. It just seems to me that after
Copernicus proposed his new system of planets going around the sun (which was
much less accurate than Ptolemy's system when Copernicus's planetary orbits
were assumed to be circles and so really was inconsistent with the data in a
way that Ptolemy's system was not), and particularly after Kepler found the
elliptical orbits, it was found that this new system was simpler and more
predictive (but only after Kepler) than Ptolemy's system, though I'm sure that
with enough epicycles Ptolemy's could have been made sufficiently accurate to
match the improved observations, at least until one got to the stage where one
needed to include gravitational interactions between the planets.
I think one other possibly relevant point is that apparently the
voyagers only measured, with an uncertainty less than one day, solar time
(given by the angle of the sun relative to the voyagers' up-down, north-south,
and east-west frame) and not sidereal time (given by motion relative to the
distant stars, such as of the moon or planets). If they had carried a table of
the times of the phases of the moon and had been able to observe them to a
precision of time shorter than one day, or similarly of the motion of any of
the planets relative to the "fixed stars," then they could have noticed a
discrepancy with the solar time that they recorded, even before talking with
the Cape Verde Islanders. The fact that they were surprised to find out that
they were missing one solar day strongly suggests that they had not considered
a possible discrepancy between their local solar time and sidereal time
(predictable in both Ptolemy's system and in any more modern one), and so they
would presumably would not have been motivated to look for one. With their
small ships and small budget, they might not have even carried a very extensive
table of astronomical data sufficient for determining the sidereal time.
(This raises a question in my mind of how accurately they could have
measured sidereal time if they had tried. I can't even tell most phases of the
moon within one day, at least with my naked eye: last night the moon looked
full to me, but my ten-year-old son told me it would not be full until today.
Presumably the precision these early voyagers must not have been too small a
fraction of a day, or else there would not have been the great impetus later to
develop accurate chronometers for determining longitude.)
Anyway, on the basis of these thoughts, my own opinion is that the
passage that Glenn quoted from William Manchester, A World Lit Only By Fire:
The Medieval Mind and the Renaissance: Portrait of an Age, (Boston:
Little, Brown & Co., 1992), p. 290-291, is either nonsense or else possibly
reports on a gaping lacuna in the thought of sixteenth-century Eurpoean men of
science:
"None of the great geographers, neither Aristotle, Ptolemy,
nor Pierre d'Ailly, had anticipated this riddle. Sixteenth-
century European men of science, as startled as Pigafetta and
Albo, toiled over their desks until they came up with what, they
unanimously agreed, was the only possible solution. Copernicus,
they concluded was right. The earth was rolling eastward,
completing a full cycle every day. Magellan and his men had been
sailing westward, against that rotation; having traversed a full
circle, the circumnavigators had gained exactly twenty-four
hours. Geocentrism - the age-old conviction that the earth was
the center of the universe - was therefore discredited. The
earth was not only round; it was moving. In fact, it was
revolving around its own axis."
So the historical question is, did these sixteenth-century European men
of science really "toil over their desks until they came up with what, they
unanimously agreed, was the only possible solution"? Did Manchester really
have any evidence for that, or did he just make it up? (I'd guess the latter,
since it would be easier for this logical oversight to be had by one person,
namely him, than by all the sixteenth-century European men of science who
supposedly toiled over the puzzle, unless possibly their number really was very
few, but even then I'd be more inclined to trust them rather than him.)
I suppose it might be interesting for someone with an historical bent
(Glenn Morton and/or Ted Davis?), if he/she/they agree(s) with my analysis and
find(s) that Manchester had no historical justification for his statements, to
write a rebuttal to Manchester's account, but maybe this is just a drop in the
bucket of Manchester's sillinesses about the "Dark Ages" if Ted is right.
I am also currious as to whether indeed "none of the great geographers,
neither Aristotle, Ptolemy, nor Pierre d'Ailly, had anticipated this riddle."
I can easily imagine that none of the crew on the voyage knew of any such
anticipation, and they could not use a search engine on the WWW to try to find
the answer when confused by the islanders, but is it true that no one knows of
any records stating that anyone in history had anticipated the "missing day"
before it occurred?
Incidentally, Glenn's nice quote from Antonio Pigafetta, First Voyage
Around the World, in Charles E. Nowell, Editor, Magellan's Voyage around the
World, Northwestern University Press, 1962, p. 256, "However, as was told us
later, it was no error, but as the voyage had been made continually toward the
west and we returned to the same place as does the sun, we had made that gain
of twenty-four hours, as is clearly seen," sounds to me slightly more natural
as if written from a viewpoint of the sun's going around the earth ("returned
to the same place as does the sun") rather than from a viewpoint of the earth's
rotating. Of course, this doesn't prove much, since we also still commonly use
the former viewpoint in speaking of sunrise, sunset, the sun rising higher in
the sky, etc., but it seems to suggest that even Pigafetta did not go to the
rotating earth viewpoint when he gave this brief resolution of the enigma.
Don Page